Inference¶
Twograph hypothesis testing¶

graspologic.inference.
latent_position_test
(A1, A2, embedding='ase', n_components=None, test_case='rotation', n_bootstraps=500)[source]¶ Twosample hypothesis test for the problem of determining whether two random dot product graphs have the same latent positions.
This test assumes that the two input graphs are vertex aligned, that is, there is a known mapping between vertices in the two graphs and the input graphs have their vertices sorted in the same order. Currently, the function only supports undirected graphs.
Read more in the tutorials
Parameters: A1, A2 : nx.Graph, nx.DiGraph, nx.MultiDiGraph, nx.MultiGraph, np.ndarray
The two graphs to run a hypothesis test on. If np.ndarray, shape must be
(n_vertices, n_vertices)
for both graphs, wheren_vertices
is the same for bothembedding : string, { 'ase' (default), 'omnibus'}
String describing the embedding method to use:
 'ase'
 Embed each graph separately using adjacency spectral embedding and use Procrustes to align the embeddings.
 'omnibus'
 Embed all graphs simultaneously using omnibus embedding.
n_components : None (default), or int
Number of embedding dimensions. If None, the optimal embedding dimensions are found by the Zhu and Godsi algorithm.
test_case : string, {'rotation' (default), 'scalarrotation', 'diagonalrotation'}
describes the exact form of the hypothesis to test when using 'ase' or 'lse' as an embedding method. Ignored if using 'omnibus'. Given two latent positions, \(X_1\) and \(X_2\), and an orthogonal rotation matrix \(R\) that minimizes \(X_1  X_2 R_F\):
 'rotation'
 \[H_o: X_1 = X_2 R\]
 'scalarrotation'
 \[H_o: X_1 = c X_2 R\]
where \(c\) is a scalar, \(c > 0\)
 'diagonalrotation'
 \[H_o: X_1 = D X_2 R\]
where \(D\) is an arbitrary diagonal matrix
n_bootstraps : int, optional (default 500)
Number of bootstrap simulations to run to generate the null distribution
Returns: p_value : float
The overall p value from the test; this is the max of 'p_value_1' and 'p_value_2'
sample_T_statistic : float
The observed difference between the embedded positions of the two input graphs after an alignment (the type of alignment depends on
test_case
)misc_stats : dictionary
A collection of other statistics obtained from the latent position test
 'p_value_1', 'p_value_2' : float
 The p value estimate from the null distributions from sample 1 and sample 2
 'null_distribution_1', 'null_distribution_2' : np.ndarray (n_bootstraps,)
 The distribution of T statistics generated under the null, using the first and and second input graph, respectively. The latent positions of each sample graph are used independently to sample random dot product graphs, so two null distributions are generated
See also
References
[1] Tang, M., A. Athreya, D. Sussman, V. Lyzinski, Y. Park, Priebe, C.E. "A Semiparametric TwoSample Hypothesis Testing Problem for Random Graphs" Journal of Computational and Graphical Statistics, Vol. 26(2), 2017

graspologic.inference.
latent_distribution_test
(A1, A2, test='dcorr', metric='euclidean', n_components=None, n_bootstraps=500, workers=1, size_correction=True, pooled=False, align_type='sign_flips', align_kws={}, input_graph=True)[source]¶ Twosample hypothesis test for the problem of determining whether two random dot product graphs have the same distributions of latent positions.
This test can operate on two graphs where there is no known matching between the vertices of the two graphs, or even when the number of vertices is different. Currently, testing is only supported for undirected graphs.
Read more in the tutorials
Parameters: A1, A2 : variable (see description of 'input_graph')
The two graphs, or their embeddings to run a hypothesis test on. Expected variable type and shape depends on input_graph attribute
test : str (default="hsic")
Backend hypothesis test to use, one of ["cca", "dcorr", "hhg", "rv", "hsic", "mgc"]. These tests are typically used for independence testing, but here they are used for a twosample hypothesis test on the latent positions of two graphs. See
hyppo.ksample.KSample
for more information.metric : str or function (default="gaussian")
Distance or a kernel metric to use, either a callable or a valid string. If a callable, then it should behave similarly to either
sklearn.metrics.pairwise_distances()
or tosklearn.metrics.pairwise.pairwise_kernels()
. If a string, then it should be either one of the keys insklearn.metrics.pairwise.PAIRED_DISTANCES
one of the keys insklearn.metrics.pairwise.PAIRWISE_KERNEL_FUNCTIONS
, or "gaussian", which will use a gaussian kernel with an adaptively selected bandwidth. It is recommended to use kernels (e.g. "gaussian") with kernelbased hsic test and distances (e.g. "euclidean") with all other tests.n_components : int or None (default=None)
Number of embedding dimensions. If None, the optimal embedding dimensions are found by the Zhu and Godsi algorithm. See
selectSVD()
for more information. This argument is ignored ifinput_graph
is False.n_bootstraps : int (default=200)
Number of bootstrap iterations for the backend hypothesis test. See
hyppo.ksample.KSample
for more information.workers : int (default=1)
Number of workers to use. If more than 1, parallelizes the code. Supply 1 to use all cores available to the Process.
size_correction : bool (default=True)
Ignored when the two graphs have the same number of vertices. The test degrades in validity as the number of vertices of the two graphs diverge from each other, unless a correction is performed.
 True
 Whenever the two graphs have different numbers of vertices, estimates the plugin estimator for the variance and uses it to correct the embedding of the larger graph.
 False
 Does not perform any modifications (not recommended).
pooled : bool (default=False)
Ignored whenever the two graphs have the same number of vertices or
size_correction
is set to False. In order to correct the adjacency spectral embedding used in the test, it is needed to estimate the variance for each of the latent position estimates in the larger graph, which requires to compute different sample moments. These moments can be computed either over the larger graph (False), or over both graphs (True). Setting it to True should not affect the behavior of the test under the null hypothesis, but it is not clear whether it has more power or less power under which alternatives. Generally not recomended, as it is untested and included for experimental purposes.align_type : str, {'sign_flips' (default), 'seedless_procrustes'} or None
Random dot product graphs have an inherent nonidentifiability, associated with their latent positions. Thus, two embeddings of different graphs may not be orthogonally aligned. Without this accounted for, two embeddings of different graphs may appear different, even if the distributions of the true latent positions are the same. There are several options in terms of how this can be addresssed:
 'sign_flips'
 A simple heuristic that flips the signs of one of the embeddings,
if the medians of the two embeddings in that dimension differ from
each other. See
graspologic.align.SignFlips
for more information on this procedure. In the limit, this is guaranteed to lead to a valid test, as long as matrix \(X^T X\), where \(X\) is the latent positions does not have repeated nonzero eigenvalues. This may, however, result in an invalid test in the finite sample case if the some eigenvalues are same or close.
 'seedless_procrustes'
 An algorithm that learns an orthogonal alignment matrix. This
procedure is slower than sign flips, but is guaranteed to yield a
valid test in the limit, and also makes the test more valid in some
finite sample cases, in which the eigenvalues are very close to
each other. See
graspologic.align.SignFlips
for more information on the procedure.
 None
 Do not use any alignment technique. This is strongly not recommended, as it may often result in a test that is not valid.
align_kws : dict
Keyword arguments for the aligner of choice, either
graspologic.align.SignFlips
orgraspologic.align.SeedlessProcrustes
, depending on thealign_type
. See respective classes for more information.input_graph : bool (default=True)
Flag whether to expect two full graphs, or the embeddings.
 True
 This function expects graphs, either as NetworkX graph objects or as adjacency matrices, provided as ndarrays of size (n, n) and (m, m). They will be embedded using adjacency spectral embeddings.
 False
 This function expects adjacency spectral embeddings of the graphs, they must be ndarrays of size (n, d) and (m, d), where d must be same. n_components attribute is ignored in this case.
Returns: p_value : float
The overall p value from the test.
sample_T_statistic : float
The observed difference between the embedded latent positions of the two input graphs.
misc_stats : dictionary
A collection of other statistics obtained from the latent position test
 null_distribution : ndarray, shape (n_bootstraps,)
 The distribution of T statistics generated under the null.
 n_components : int
 Number of embedding dimensions.
 Q : array, size (d, d)
 Final orthogonal matrix, used to modify
X
.
References
[1] Tang, M., Athreya, A., Sussman, D. L., Lyzinski, V., & Priebe, C. E. (2017). "A nonparametric twosample hypothesis testing problem for random graphs." Bernoulli, 23(3), 15991630. [2] Panda, S., Palaniappan, S., Xiong, J., Bridgeford, E., Mehta, R., Shen, C., & Vogelstein, J. (2019). "hyppo: A Comprehensive Multivariate Hypothesis Testing Python Package." arXiv:1907.02088. [3] Alyakin, A. A., Agterberg, J., Helm, H. S., Priebe, C. E. (2020). "Correcting a Nonparametric Twosample Graph Hypothesis Test for Graphs with Different Numbers of Vertices" arXiv:2008.09434